Techniques are presented for determining the MHD equilibrium configuration for circular and non-circular plasmas from measurements of the magnetic field and flux made outside the discharge. Least-squares minimization techniques are introduced to provide an objective means for comparing the experimental data with the corresponding calculated equilibrium and for error estimation. Free-boundary equilibria using a simple model for the plasma-current profile with three free parameters – total current, poloidal beta, and a single parameter describing the peakedness of the current profile – are found to be statistically adequate and sufficient to describe the magnetic configuration of the discharge. In plasmas with significant non-circularity, the Shafranov dimensionless inductance, ℓi, can be determined independent of βp. Examples of the application of the technique to discharges in the Doublet-Ill tokamak are presented.